I'm going to my cousins and need to sign my account into his Xbox 360.
I have Xbox One and he has Xbox 360. I want to download Minecraft, from my account, on to his Xbox. Can this be done?
You can sign into the same Xbox account, yes. Both Xbox 360 and Xbox One use the same Xbox account to log players into Live.
However, unless you own an Xbox 360 version of Minecraft, you will not be able to download the game, as Xbox One games are not compatible with Xbox 360. You won't even be able to access the games list from the store.
There is no such thing as an Xbox One account. The account you use to log into your Xbox One is your Live account. This account is used (list not exhaustive) for:
Once you signed into your account you can use all the features and products you own for the particular platform. So if you own Minecraft for the 360, you can sign in and download it to your friend's console. However, the game is tied to your account and your friend would need your account to be logged onto his console to play the game.